Abstract: The LUXE experiment will investigate the strong-field QED regime by using the interactions of high-energy electrons from the European XFEL in a powerful laser field. It will measure the production of electron-positron pairs as a function of the laser field strength, up to the non-perturbative non-linear regime. LUXE foresees a positron detection system consisting of a tracker and a granular and unprecedentedly compact silicon-tungsten electromagnetic sandwich calorimeter (ECAL-P). The ECAL-P has been designed to cope with the wide range of the expected number of positrons per bunch crossing. In addition, the energy distribution of the positrons has to be measured on top of a widely spread low-energy background. The ECAL-P is composed of tungsten absorber plates interspersed with thin sensor planes, consisting of silicon pad sensors, flexible Kapton printed circuit planes, and carbon fiber support. The sensor planes are less than 1 mm thick and will be read using dedicated front-end ASICs in 130 nm technology (FLAXE) and FPGAs for data pre-processing. GaAs sensor planes with integrated readout strips are also being considered as an alternative to silicon. Prototypes of individual sensor planes have been tested in a 5 GeV electron beam. A full compact calorimeter tower of up to 90 x 90 x 600 mm3 (15 X0) will be produced and tested in an electron beam. The design challenges, sensor characterization, prototyping, integration, commissioning, and the available results from a beam test in 2022 are discussed.